What makes two colors, or two shapes, or two words in a poem compatible? It is an artistic choice. It is the same with chords. A style is what may provide some guidance. E.g. a contemporary pop or rock chord progression maybe wouldn't sound good in a classical composition. That being said, classical harmony is quite well described and codified, but it's also a quite broad topic, difficult to summarize in a short answer.

I’m voting to close this question because it hovers between aesthetics and physics, and the entire theory of Tonality is intended as one answer to this, but is too much to explain in a brief Q&A.

This is just one reason why music theory is only that - theory. There is no 'law' or 'rule' which can justify compatible chords.

Music theories are more or less formal or systematic attempts to describe musical practices. Is this question asking for an overview of all different kinds of description attempts of all kinds of harmonic aspects of all kinds of musical practices anywhere ever, or just some specific kinds of descriptions of some specific kinds of musical practices? Maybe it's limited to those practices where a concept of "chord" has been used. If the OP could give a few examples of "explanations" of "compatibility" and whose musical practices those explanations apply to. Please.

@Tim Isn't there a theory describing why some chords are in harmony with each other?

"what's the difference": consider an example. It's common to have a chord sequence such as F-G-C, so F and G should presumably be considered compatible if played sequentially. But if you play them together, you have six pitches sounding simultaneously, FGABCD, which is an unusual sonority and would (in most theoretical systems) lead to the statement that F and G are not compatible to play simultaneously.

@piiperiReinstateMonica Of course, I understand that different musical styles have different combinations of chords and often the principles of playing, but I am interested in how the harmony of some chords with each other is theoretically explained (it does not matter two chords or a large sequence).

@phoog thanks for the explanation, but did you say "in most theoretical systems"? Are we talking about microtones or maybe just musical styles? (jazz, flamenco, etc.)

Even the "line between bad and normal" depends on the context. Major 7ths and minor 9ths are normally overly dissonant intervals, but I've heard 7b9 chords and chords shaped like E-G#-D-G too often in notated music to ignore.

Any kind of explanation is theoretical. There is no such thing as non-theoretical explanation. All descriptions of musical practices are theory. What should be described as good or bad or whatever for some purpose, depends on the context and purpose. For what purpose and what audience are you targetting the music where you intend to use chords, and how can you tell if a chord is "compatible" with another? Or are you asking for opinions from others - what does person X consider "compatible" for purpose Y in context Z, for all (X, Y, Z)?

I just think this question is too broad. It’s almost like the answer would be explaining more than 50% of all music theory and would include different answers for different genres and cultures. It’s just too much. A similar question that we could answer would be “why do these two particular chords go together in this particular song?”

Now we know you're playing 2 chords simultaneously, the question's still no easier to answer. There are cases - Cmaj/Em where it makes Cmaj7, or G maj/Bo, making G7, but most times the result will be a horrible discord, in most styles of music. in fact, answering is impossible!

@piiperiReinstateMonica it's just that for me, a "theoretical explanation" is still an explanation that refers to an existing theory, since I often meet people who answer something like this: "well, um, I had such a situation and therefore I think like this" (that is, simply referring to personal experience, etc.)

@Tim So I don't need to take into account which style of music is good and which is bad. I edited the question where I pointed out that it is quite simple without a contextual situation - just a certain number of chords are played at the same time. What do I need to study in the field of solfeggio in order to understand in which case these chords are harmonious with each other and in which they are not? (not by ear, but exactly the rules)

Re the original question. Music theory considers the terms "consonance" and "dissonance" rather than good and bad or compatibility or similar. Loosely, consonance means, "Could stop here," and dissonance means, "Ought to move." The concepts can apply to intervals, chords, key areas, movements,.... It's recursive. Perhaps a reformulation here could help. There are some discussions in outline form on theory pages.

@ToddWilcox Once again, I do not need to take into account the context, this is not a moment in the composition, but I just want to know what rules are responsible for whether the chords played at the same time are harmonious. If there are such a huge number of these rules that they occupy 50% of musical theory, then I ask you to just tell me where I can study them or how to Google them correctly at least.

@ttw Maybe this is what I'm looking for? "Consonance is a harmonious combination of several sounds" - are there rules for building consonances? (sorry, I really can't find it)

You need to provide some kind of style context. Chord can sound "compatible" in music by someone like Darius Milhaud who used polychords or the Renaissance music of someone like John Bull, but you would explain that compatibility differently.

In case it isn't clear by now, the core problem is that you're looking for a scientific, algorithmic approach to an aesthetic phenomenon. Even if there are simple answers—a C chord with a C# chord will "sound bad" because of minor 2nds—then there are problems when you try to extrapolate rules (add a single minor 2nd and it might "sound good"), and other variables matter (put one of the chords very low and the other high and it might not "sound bad"). The trouble is that "sound good" is itself not scientifically measurable.

However, even if you can't have algorithms, you can have vague, general guidelines. No, I don't think anyone would stack a C chord and a C# chord unless they were explicitly looking for dissonance. So stick with ttw's answer: treat 2nds, 7ths, and tritones with suspicion.

“If there are such a huge number of these rules that they occupy 50% of musical theory, then I ask you to just tell me where I can study them” there are no rules but here is a good place to start studying theory: viva.pressbooks.pub/openmusictheory

@AndyBonner A C below a C# chord could work as a C11(b9,b13) in the right context. Not sure if that’s the right symbol but jazz makes many things possible

@ToddWilcox Since the OP is asking about it, I'm talking about "chord plus chord," not just "note added to chord." It's kind of an odd frame of reference though because we don't normally superimpose one chord with another, except to then call the result by a special name. I'd venture that 90% of the music commonly consumed is "melody notes over a single given chord."